Euclidean path integral

1. argonurbawono

18
i am learning path integral for quantum field theory, and my professor used euclidean time (imaginary time) and most textbooks use minkowski time.

does actually changing the time from real (minkowski) into euclidean (imaginary) CHANGE the physics in some way?

8,147
Staff Emeritus
No, because path integrals aren't about describing physics but about calculating number FROM the existing physics.

What your professor is doing is called Wick rotation, and is motivated like this. The raw PI calculation uses time (in Minkowski space) that can be described as a point on the real line, but the raw calculation doesn't converge because the factors like $$e^{-i\alpha t}$$ define sines and cosines which just oscillate as t gets big. Wick said, look at the ral line as just the real axis of the complex plain. Those exponentials are analytic, so we can do an analytical continuation over the first quadrant (with an arbitrarily large quarter circle to complete it) and evalauate the same integrals with the imaginary axis of that quadrant. In effect you have rotated the real axist through 90 degrees. And the exponentis are now negative real and the exponentials decay at large t.

Thus you have the $$\tau = i t$$ change of variables that your Professor showed you and whan you have completed the calculation you can "rotate" back to Minkowski time and everything's copacetic.

3. argonurbawono

18
fine.
but what about the case of instantons? when you have potential of the form $$V(x) = \frac{g^2}{8}(x^2-a^2)^2$$, the potential looks different in euclidean and minkowski.

in minkowski you have two wells and classically the particle is either on the right or on the left. but in euclidean, the potential gets inverted and we end up with a well in the middle, suggesting a classical particle to oscillate in the middle, which corresponds to oscillating inside the barrier in the minskowski. remember the kinetic term becomes negative in euclidean because of x dot square term, and the corresponding energy inverts the orientation of the potential if we solve for position for ground state.

this (i think) change the picture of the physics, since we change the position of the minima. don't you think?

4. Haelfix

1,748
Wick rotation is technically valid (with strong caveats regarding the existance of suitable analytic continuations) in perturbation theory.. Indeed there are known exact solutions of the path integral in 2d that does not respect wick rotation. Instantons are nonperturbative kinks that path integral based perturbation series misses order by order so you have be careful about what you are doing.

On the other hand, when its allowed, wick rotation can turn a horribly illdefined and pathological beast (the minkowski path integral) into something much more well behaved and physical.

The exact mathematics behind this whole operation is still very unclear and an open problem (and indeed part of a problem for the Clay institute)